This post is motivated by recent discussionsinvolving a term with which I had beenpersonally unfamiliar. It is an exercisein "basic" logic.

Consider the "math trick,"

9.999...-0.999...--------- 9.000...

in relation to

10x -x---- 9x

By pattern matching, one infers

0.999...=1

x=1

Since x=1 is surmised from the algebraicsyntax.

Now, in what follows, the particularproblem will be considering the natureof eventually constant sequences takento be ontologically "the same." So, inaddition to the above, the identity,

1=1.000...

is considered to be a presuppositionavailable to the analysis. So, whatis actually being considered is thesyntax,

0.999...=1.000...

and, at the appropriate time, theanalysis will turn to binary sequencesfor ease of presentation. Thus, this analysisis not necessarily applicable to theexample with which it has been introduced.

It will be, however, applicable to thecontext from which the term "distinguishability"has been found to have a sourced definition.

In the tradition of logic that has becomepopular, the logical problem being investigatedmay be stated as being an assertion of ontologicalidentity using an informative identity statement. Thatproblem is particularly difficult, in general, asthe classic Fregean identity puzzle

"Hesperus is Phosphorus"

has shown. But, by analogy with polynomialsof degree greater than 4, there is no reasonto think that some classes of problems ofthis nature are not amenable to analysis.

To be formally explicit, the analysis presentedhere is based on Sausserian semiotics wherein asyntactic signal antecedent to its consequentuse as a sign never regresses beyond anobservable sign-vehicle (i.e., an inscription).In contrast, Piercean semiotics is subjectto infinitary regress into non-grammaticalsign-functionality.

So, that we clarify the nature of the receivedparadigm in this matter, we address the issueof uniform semantic interpretation of inscriptionsby invoking Carnap's notion of syntactic equality.Hence, what is expressed by

0.999...=1.000...

is the identity of two equivalence classes

[0.999...]=[1.000...]

relative to which some quotient model mustbe formed to accommodate the fact that anontological assertion is being made usingan informative identity.

Because the received paradigm does notaddress informative identity directly, itis possible that there are interpretationsin which

[0.999...]=[1.000...]

is false.

With these preliminaries out of the way,it is time to consider how the presupposition

1=1.000...

is indicated as the canonical representationgiven that, alternatively,

1=0.999...

is also a candidate for the canonicalrepresentation.

Consider the construction of the realnumbers in relation to Dedekind cuts. Toaccept this construction is to accept thefact that in a hierarchy of definition, thereis information about the defined system thatis epistemically prior by virtue of theconstruction.

What is taken to be known essentially is thatfor any given Dedekind cut, it is in the classof rational cuts or it is in the class of irrationalcuts.

However, there is still more information associatedwith the construction that informs with respect tothe analysis at hand. Because one has a choice asto which direction of the linear order in the underlyingrationals will be used for the cuts, there is adecision that may be applied to the representationsof rationals in relation to decimal expansions.

Since, for any finite initial segement of theexpression 0.999... it is held that

0.999 < 1.000...

the system of Dedekind cuts will be given itsfundamental partition according to whether ornot its greatest lower bound is an element. Thischoice is compatible with the usual notion thatan eventually constant sequence of trailing zerosis the same as a long division that terminates.

It is at this point that the context of theanalysis will be changed to binary strings. Theprincipal concern will be the relationship ofeventually constant strings to one another inconstrast to strings that never become eventuallyconstant.

The next step is to represent the orientation ofcuts with respect to a preferred canonicalrepresentation.

Now, what motivates the choice of theoreticalcontext for this analysis is the informationcontent in the assertion,

0.999...=1.000...

The model that best seems to reflect a selectionof preference here is that of automata -- inparticular, the selection of preference ischaracterized in terms of lossless and lossymachines.

Since it is almost trivial, we begin by recallingthat there is prior knowledge that distinguishesthe cuts into two classes and look to those cutsthat are defined as irrational real numbers. Letthe automaton that is applied to identifyingconcatenations of alphabet letters for the irrationalcuts simply copy its input to its output,

Thus, each state S_k copies its input symbol toits output symbol and transitions to its grammaticalsuccessor S_(k+1)

The simplicity of the irrational automaton arisesfrom the fact that it is applied to the identifyingsequence for a given Dedekind cut when that cut isnot a rational cut. It is defined negatively, and,the simplicity of its content reflects that fact.

For the rational numbers, the situation is differentbecause the hierarchy of logical definition affordsprior knowledge. The automaton for each rationalcut simply outputs the identifying concatenation ofalphabet letters as a known sequence, regardlessof input symbols. Those rational numbers that do not enjoytwo distinct representations by virtue of eventuallyconstant sequences are lossless. That is,

Although each state transitions to itsgrammatical successor as with the machineapplied to irrational numbers, the outputsymbols reflect the state of prior knowledgeand are not simply copies of the inputsymbols.

In constrast, those rational cuts with aplural multiplicity of identifying concatenationsof alphabet letters have a different patternof state transitions from the prior cases.

Now, the following quoted material is thedefinition for distinguishability that shallbe considered here. It is taken from "Finite-StateModels for Logical Machines" by Frederick C. Hennie.Although it is not the best, it is all that is onmy bookshelves. To see what I mean in this regard,consider the subsequent description for the associatedpartitions carefully. By my reading, the uqualifieddescription for the construction of partitions wouldnot partition the states in the manner suggested bythe subsequent qualifying remarks describing asequence of refinements. Perhaps I am wrong.Sometimes mathematics is so easy that it is hard.I included the qualifying remarks concerning thedescription of partitions specifically because Iam having a hard time seeing the claim directlyfrom the unqualified statement.

With the qualifying remarks, the description ofthe partitions appears to correspond with thedefinition.

Distinguishability is defined as follows

"The most convenient way of finding equivalencesthat exist among the states of a machine is toconcentrate on the states that are not equivalent.Thus, we say that two states are *distinguishable*iff there exists a finite input sequence thatyields one output sequence when the machine isstarted in one state and a different output sequencewhen the machine is started in the other state.If this input sequence contains k symbols, thestates are said to be distinguishable by an experimentof length k, or simply k-distinguishable. Statesthat are not distinguishable by any experiment oflength k or less are called k-equivalent. Itfollows that two states are equivalent iff theyare k-equivalent for every finite value of k.

"The definition of k-distinguishability becomesmore useful when recast in the form of tworules:

(1) Two states are distinguishable by an experimentof length one iff there is some input symbol thatproduces different output symbols according towhich of the two states the machine is in.

(2) Two states are distinguishable by an experimentof length k (with k>1) iff there is some input symbol,say z, such that the z-successors of the two statesare distinguishable by an experiment of length (k-1).

These rules provide the basis for a step-by-stepprocedure for determining which states of a machineare one-equivalent, which are two-equivalent, andso on.

"The first step is the formation of a partition P_1in which two states are placed in the same block iffthey produce identical output symbols for each possibleinput symbol. This clearly puts two states in thesame block of P_1 if they are one-equivalent and indifferent blocks if they are one-distinguishable."

For the automaton applied when the cut correspondsto an irrational,

P_1=(A,B,C,D,...)

For that applied to the singly-represented rationalthe example state table yields,

P_1=(A,B,E,F,...)(C,D,G,H,...)

For the doubly-represented rational example, thisis the partition before a canonical representativeis indicated through the Dedekind cut,

"The next step is the formation of a partitionP_2 in which two states are placed in the sameblock iff, for each input symbol z, their z-successorslie in a common block of P_1. Note that P_2must be a refinement of P_1, for if two statesare two-equivalent, they must certainly beone-equivalent. Partition P_2 is mosteasily formed by splitting the various blocksof P_1 in such a way that the definingconditions for P_2 are met."

For the automaton applied when the cut correspondsto an irrational, the second partition is given as

P_2=(A,B,C,D,...)

Observe that there has been no change.

For that applied for the singly-represented rationalthe second partition for the example state tableyields,

P_2=(A,E,...)(B,F,...)(C,G,...)(D,H,...)

For the doubly-represented rational example, thisis the second partition before a canonical representativeis indicated through the Dedekind cut,

"Partitions P_3, P_4, ..., P_k,... can be formedin a similar manner.[...]

"[...], we see that the partitioning processmay be terminated as soon as some partitionP_(m+1) is found to be identical to itspredecessor P_m."

Thus, the automaton applied when the cut correspondsto an irrational has already terminated itssequence of partitions given

P_1=(A,B,C,D,...)

P_2=(A,B,C,D,...)

For that applied for the singly-represented rationalthe partition sequence for the example state tableyields,

P_1=(A,B,E,F,...)(C,D,G,H,...)

P_2=(A,E,...)(B,F,...)(C,G,...)(D,H,...)

P_3=(A,E,...)(B,F,...)(C,G,...)(D,H,...)

For the doubly-represented rational example beforeorientation by the Dedekind cut,

P_1=(A,B,D,G,I,K,...)(C,E,F,H,J,L,...)

P_2=(A,G,I,K,...)(B,D)(C)(E)(F,H,J,L,...)

P_3=(A)(B)(C)(D)(E)(F,H,J,L,...)(G,I,K,...)

P_4=(A)(B)(C)(D)(E)(F,H,J,L,...)(G,I,K,...)

and after,

P_1=(A,B,D,E)(C,F,G,H,I,J,K,L,...)

P_2=(A)(B,D,E)(C)(F,G,H,I,J,K,L,...)

P_3=(A)(B)(C)(D,E)(F,G,H,I,J,K,L,...)

P_4=(A)(B)(C)(D,E)(F,G,H,I,J,K,L,...)

So, up to this point, it is clear that the variousrelationships in the definition of real numbersvia Dedekind cuts and the presuppositions ofa trivial algebraic substitution do have discerniblerepresentation relative to modeling with automatons.

With regard to the relationship of finitism andthe apparent simplicity of a symbol such as

1.000...

it is instructive to look at the reduced machines forthese examples. For the automaton applied when thecut is an irrational, one has

| 0 | 1 |--|----|----|A | A0 | A1 |--|----|----|

For that applied when the cut corresponds with asingly-represented rational, one has

In all cases, the finiteness of the reduced representationrelies on circularity. In the cases where the cuts areeither irrational or doubly-represented, one or more statesare found to be transitioning into themselves.

Relative to the model investigated here, the notionof reduced machine relative to k-equivalent statespermits all of the possible machine configurationsderived from consideration of Dedekind cuts to havefinite representation -- provided one does not objectto the use of circularity in the state tables.

The problem with simply writing out sequencesand naively "knowing" the intended meaning of"distinguishability" is that the automaton forthe irrational cuts is applied to the identifyingconcatenation of alphabet for all sequenceswithout regard for the hierarchy of logicaldefinition.

This analysis, as far as it goes, is not yetadequate.

The definition of distinguishability is characterizedin terms of "experiments." It seems prudent toconsider this aspect of the definition further andwith respect to the infinitary nature of the originalproblem.

For each state, there are two possible outcomes inthe sense of traversing a decision tree. Supposestates are redefined as choices and the componentsof each state are taken to be selections. Thenan experiment is a sequence of selections takenthe purpose of destroying an assertion of equivalence.

In other words, just a subtle change of languagecharacterizes this model in relation to the historicalepistemic condition associated with the assertionof "sameness" in relation to definitions.

However, the primary purpose for this further analysisis to construct a topology.

One thing that will be required for this constructionis an enumeration of fractions which is the familiardiagonalized listing on Z^+xZ^+.

Each fraction, representing a rational number,corresponds to one of the automata discussed inthe preceding remarks. As before, there is noimmediate concern for uniqueness constraints sincethat destroys information.

Given any fraction, take the reduced state table, say M,for the lossless representation and organize the selectionsfor the unreduced state table relative to their grammaticalorder according to

A_0---B_0---C_0---...M...---C_1---B_1---A_1

Let this ordering be given the interval topology.Observe that M is the endpoint of no interval.

Now, let the enumeration of fractions be given thediscrete topology, and, as a first specificationfor the topology being constructed, let an enumerationof sequences like that above be given the producttopology.

For the next step, it is easiest to use numericalcoordinates. So, let the machine symbol M betaken as set-theoretic omega and let the selectionsbe indexed by the non-zero integers according to

(1)---(2)---(3)---...omega...---(-3)---(-2)---(-1)

Extend the construction with the letters of theinput alphabet, namely '0' and '1'. Rememberingthat the interval topology on the representationabove has, for example, (2)<omega and omega<(-2),and, remembering that the omega in the representationabove has an natural number index according to theenumeration of fractions, augment the topologydescribed so far with basis elements,

B_n_0={0}u{(i,j)|i<omega /\ j>=n}

B_n_1={1}u{(i,j)|i>omega /\ j>=n}

This is the minimal Hausdorff topology.

Before continuing with the present example, take a momentto consider a countable language with a unary negationsymbol.The well-formed formulae may be partitionedaccording to whether or not the first symbol fromamong the logical constants is the symbol for negation.They can be partitioned into an enumerable sequenceof lines according to the formulae that can be formedrelative to each step of a stepwise introduction of variableterms. The negation symbol, itself can be taken as"omega." Then, the formulae can be arranged in this mannerrelative to extension with the Fregean constants,"the True" and "the False." Language, at least thatfragment that can be formalized, is topological.

Returning to the present analysis, consider thequotient topology induced by a map of this minimalHausdorff topology into the set of reduced losslessmachines applied to the identifying sequences ofalphabet letters for rational Dedekind cuts. Sucha topology exists because the map from the minimalHausdorff topology is surjective onto the set oflossless machines provided that the map takes each"line" of selections to the reduced machine for whichits choices form the states of the unreduced machine.

Because the topology on the enumeration of fractionsis discrete, the inverse image of each machinecorresponds to a representation of the equivalenceclass of fractions for some particular rationalDedekind cut.

Based on the inverse images, there is a partitionof the minimal Hausdorff topology by which a quotientspace on the topology may be formed.

Something for another time.

In any case, the definition of "distinguishability" isobtained by a stepwise condition involving "experimentsof length K". But -- and here is the important aspectof this analysis -- equivalence is an infinitary concept.You do not get to write

0.999...=1.000...

with neither circularity nor a completed infinity.It is one or the other.